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ALGORITHMICA

2011

2011

We show that computing the crossing number and the odd crossing number of a graph with a given rotation system is NP-complete. As a consequence we can show that many of the well-known crossing number notions are NP-complete even if restricted to cubic graphs (with or without rotation system). In particular, we can show that Tutte’s independent odd crossing number is NP-complete, and we obtain a new and simpler proof of Hliněný’s result that computing the crossing number of a cubic graph is NP-complete. We also consider the special case of multigraphs with rotation systems on a ﬁxed number k of vertices. For k = 1 we give an O(m log m) algorithm, where m is the number of edges, and for loopless multigraphs on 2 vertices we present a linear time 2-approximation algorithm. In both cases there are interesting connections to edit-distance problems on (cyclic) strings. For larger k we show how to approximate the crossing number to within a factor of k+4 4 /5 in time O(mk log m) on a...

Related Content

Added |
12 May 2011 |

Updated |
12 May 2011 |

Type |
Journal |

Year |
2011 |

Where |
ALGORITHMICA |

Authors |
Michael J. Pelsmajer, Marcus Schaefer, Daniel Stefankovic |

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